Okay, so the way a virus (any virus) works is that a little bit enters your body. It starts to multiply, spreading as it does so like mold on bread. The virus levels increase until there's enough in your body to make you ill. Your immune system kicks in, fighting the virus to protect you, and you get better.
For covid, two or three days after entering your body it's multiplied to the point that's there's enough virus in your nose to show up on a PCR test.
A day or two after that, there's enough virus in your system that you blow it out whenever you sneeze, cough, or breathe. That means you can give it to other people.
A couple of days after that, there's enough virus in your body to make you feel rubbish.
And a week or so after that, if you're really unlucky, you get very sick indeed.
The vaccine means that your immune system responds faster to the virus infection. So, instead of your immune system not getting on top of it til you've felt lousy for a few days, it might do so before you get any symptoms at all. Or before there's enough virus in you that you can spread it. Or maybe before you can pick it up on a test.
It doesn't stop you from getting infected, because the immune system can't kick in until the virus is there. But it kicks in faster, stopping the infection in its tracks sooner in the progression of the disease.